A singular perturbation free boundary problem for elliptic equations in divergence form

Abstract

In this paper we study the free boundary problem arising as a limit as ε → 0 of the singular perturbation problem div(A(x)∇u) = (x)β ε (u), where A = A(x) is Holder continuous, β ε converges to the Dirac delta δ 0 . By studying some suitable level sets of u ε , uniform geometric properties are obtained and show to hold for the free boundary of the limit function. A detailed analysis of the free boundary condition is also done. At last, using very recent results of Salsa and Ferrari, we prove that if A and are Lipschitz continuous, the free boundary is a C 1,γ surface around H N−1 a.e. point on the free boundary.

Existence and regularity properties of non-isotropic singular elliptic equations

“…(14) and (15) and satisfies (7)-(9), (12) and (13). Then, for all subdomain ⊂⊂ we have Hausdorff measure bound…”

“…For the proof of the following result, we combine compactness results with an interpolation argument adapted from [17] (see also [14]). …”

“…Theorem 1.5 Assume f satisfies (3)-(6), (14). Let (u β ) 0<β<1 be the family of solutions of (1) obtained in Theorem 1.1 with ≡ 1.…”

“…Theorem 1.6 Suppose f satisfies (3)-(6), (14) and (15) and satisfies (7)-(9), (12) and (13). Let (u β ) 0<β<1 be a family of solutions of (1) obtained in Theorem 1.1.…”

“…, (14) and (15) and satisfies (7)-(9), (12) and (13). Let u = u λ be the solution of (1) provided by Theorem 1.1 and x 0 ∈ ∂{u > 0} with B r (x 0 ) ⊂ for some r > 0.…”